Dynamic Exoplanets Alexander James Mustill
Exoplanets: not (all) like the Solar System
Exoplanets: not (all) like the Solar System Solar System Lissauer et al 14
Key questions to bear in mind What is role of formation versus role of later evolution? How prevalent is instability, and what triggers it? How are outer systems and inner systems linked?
The Keplerian orbit eccentricity e=0 periapsis q=a(1-e) semi-major axis a eccentricity e~1 Image credit: Wikipedia
The Keplerian orbit Given bodies masses, orbit uniquely determined by 6 orbital elements: semi-major axis a eccentricity e inclination i true anomaly ν, f, or mean anomaly M argument of periapsis ω longitude of ascending node Ω
The Keplerian orbit In the 2-body problem, all the orbital elements are constant except for the mean anomaly (or equivalent) In a Hamiltonian framework, can use M, ω, Ω as canonical positions; conjugate momenta are then L = µ* (µa) G = µ* (µa(1-e 2 )) H = µ* (µa(1-e 2 )) cos i Hamiltonian itself is H = - µ 2 µ* 3 /(2L 2 )
The Keplerian orbit In the 2-body problem, all the orbital elements are constant except for the mean anomaly (or equivalent) In a Hamiltonian framework, can use M, ω, Ω as canonical positions; conjugate momenta are then L = µ* (µa) G = µ* (µa(1-e 2 )) Angular momentum H = µ* (µa(1-e 2 )) cos i Hamiltonian itself is H = - µ 2 µ* 3 /(2L 2 ) Energy
The N>2-body problem In general intractable 2 approaches: numerical, or analytical with simplifying assumptions For planetary dynamics, can usually treat the planets as on perturbed Keplerian orbits H = - µ 2 µ* 3 /(2L 2 ) + R(M i,ω i,ω i,l i,g i,h i ) Hill sphere where planet s gravity dominates over star s is of radius r H = a (m pl /(3m )) ~ 0.01a for Earth lots of orbits with small perturbations can mean big effects Textbook by Murray & Dermott (1999, CUP) Solar System Dynamics
Secular evolution Mutual perturbations averaged over orbital time-scales Exchange of angular momentum with no change in energy Jupiter Saturn Uranus Neptune
Kozai perturbations Achieve very high e starting from high i Particularly important in systems with wide binary companion i
Resonances Occur when two important frequencies are commensurate (e.g. 2:1) Mean motion (e.g. Neptune Pluto, asteroid belt) Secular (e.g. asteroid belt) ν6 secular resonance Drivers of instability but can be stabilising Image credit: Wikipedia
Resonances 9:8 Δa Mustill & Wyatt 12 8:7
Scattering When the perturbed Keplerian approximation breaks down: impulsive changes to orbits May occur after long period of gentler evolution Outcome governed by Safronov number: ratio of escape to orbital velocity Θ 2 = (mpl/m )(a/rpl) Leads to collision between planets or large change to orbit (leading to ejection or collision with star)
Hot Jupiters and planetary multiplicity Hot Jupiters Lissauer et al 14
Dynamical delivery Kozai, secular, scattering in outer system Followed by tidal damping Fabrycky & Tremaine 07
What happens to inner system? Take a 3-planet Kepler system as an example Throw in Jupiter without worrying about outer system (Mustill, Davies & Johansen 2015) Inner system totally cleared or Jupiter ejected
What happens to inner system? Added eccentric Jupiter Kepler-18 super Earth/Neptunes Mustill, Davies & Johansen 15
Singles in general Lissauer et al 14
Singles in general Singles in general: too many to have arisen from highermultiplicity population (Johansen et al 2012) Is this primordial (lots of systems form only singles) or a result of later evolution? Kepler triples are not unstable (Johansen et al et al 2012) Can we disrupt triples externally? What about higher multiples (Pu & Wu 2015, Gladman & Volk 2015, Mustill et al in prep)
Realistic Jupiter delivery Now treat some realistic delivery scenarios (Mustill, Davies & Johansen in prep): Kozai by binary Scattering in multi-giant system (see also Daniel Carrera in prep: effects of scattering on Earth-like planets)
Example of scattering 2 Jupiters, 2 Neptunes Kepler-18 super Earth/Neptunes a, q, Q [au] time [yr]
Example of Kozai Binary star Jupiter Kepler-18 super Earth/Neptunes a, q, Q [au] time [yr]
Some example stats system 0 planets 1 planet 2 planets 3 planets K18 + Kozaied Jupiter 44 69% 1 2% 10 16% 9 14% K18 + Kozaied Jupiter + GR 27 42% 23 36% 8 13% 6 9% K18 + scattering giants at 3au 14 11% 8 6% 4 3% 102 80% K18 + scattering giants at 1au 9 28% 4 13% 3 9% 15 47%
Some example stats system 0 planets 1 planet 2 planets 3 planets K18 + Kozaied Jupiter 44 69% 1 2% 10 16% 9 14% K18 + Kozaied Jupiter + GR 27 42% 23 36% 8 13% 6 9% K18 + scattering giants at 3au 14 11% 8 6% Best at making singles 4 3% 102 80% K18 + scattering giants at 1au 9 28% 4 13% 3 9% 15 47%
Example Quintuple a, q, Q [au] time [yr]
Quintuple stats Starting from an unstable quintuple, we end up with doubles or triples But we assume perfect merging in collisions not necessarily realistic Collision velocities ~e vkep can be significantly higher than escape velocity collisions erosive or destructive
Ongoing work Population synthesis studies of Kozai and scattering in outer systems, with Kepler systems on the inside Use higher-multiplicity Kepler systems as the templates: what happens to the multiplicities of unstable quintuples? Investigate role of non-merging collisions
Key points The N-body problem is considerably harder, and richer, than the 2-body problem Instability often preceded by long periods of more quiescent evolution Many extra-solar systems unlike our own eccentric giants close-in Super-Earths Multiplicities shaped by dynamical evolution